MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ixxss1 Structured version   Unicode version

Theorem ixxss1 11428
Description: Subset relationship for intervals of extended reals. (Contributed by Mario Carneiro, 3-Nov-2013.) (Revised by Mario Carneiro, 28-Apr-2015.)
Hypotheses
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
ixxss1.2  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z S y ) } )
ixxss1.3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W B  /\  B T w )  ->  A R w ) )
Assertion
Ref Expression
ixxss1  |-  ( ( A  e.  RR*  /\  A W B )  ->  ( B P C )  C_  ( A O C ) )
Distinct variable groups:    x, w, y, z, A    w, C, x, y, z    w, O   
w, B, x, y, z    w, P    x, R, y, z    x, S, y, z    x, T, y, z    w, W
Allowed substitution hints:    P( x, y, z)    R( w)    S( w)    T( w)    O( x, y, z)    W( x, y, z)

Proof of Theorem ixxss1
StepHypRef Expression
1 ixxss1.2 . . . . . . . 8  |-  P  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x T z  /\  z S y ) } )
21elixx3g 11423 . . . . . . 7  |-  ( w  e.  ( B P C )  <->  ( ( B  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* )  /\  ( B T w  /\  w S C ) ) )
32simplbi 460 . . . . . 6  |-  ( w  e.  ( B P C )  ->  ( B  e.  RR*  /\  C  e.  RR*  /\  w  e. 
RR* ) )
43adantl 466 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( B  e.  RR*  /\  C  e. 
RR*  /\  w  e.  RR* ) )
54simp3d 1002 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  w  e.  RR* )
6 simplr 754 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  A W B )
72simprbi 464 . . . . . . 7  |-  ( w  e.  ( B P C )  ->  ( B T w  /\  w S C ) )
87adantl 466 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( B T w  /\  w S C ) )
98simpld 459 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  B T w )
10 simpll 753 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  A  e.  RR* )
114simp1d 1000 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  B  e.  RR* )
12 ixxss1.3 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  w  e. 
RR* )  ->  (
( A W B  /\  B T w )  ->  A R w ) )
1310, 11, 5, 12syl3anc 1219 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( ( A W B  /\  B T w )  ->  A R w ) )
146, 9, 13mp2and 679 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  A R w )
158simprd 463 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  w S C )
164simp2d 1001 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  C  e.  RR* )
17 ixx.1 . . . . . 6  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
1817elixx1 11419 . . . . 5  |-  ( ( A  e.  RR*  /\  C  e.  RR* )  ->  (
w  e.  ( A O C )  <->  ( w  e.  RR*  /\  A R w  /\  w S C ) ) )
1910, 16, 18syl2anc 661 . . . 4  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  ( w  e.  ( A O C )  <->  ( w  e. 
RR*  /\  A R w  /\  w S C ) ) )
205, 14, 15, 19mpbir3and 1171 . . 3  |-  ( ( ( A  e.  RR*  /\  A W B )  /\  w  e.  ( B P C ) )  ->  w  e.  ( A O C ) )
2120ex 434 . 2  |-  ( ( A  e.  RR*  /\  A W B )  ->  (
w  e.  ( B P C )  ->  w  e.  ( A O C ) ) )
2221ssrdv 3469 1  |-  ( ( A  e.  RR*  /\  A W B )  ->  ( B P C )  C_  ( A O C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758   {crab 2802    C_ wss 3435   class class class wbr 4399  (class class class)co 6199    |-> cmpt2 6201   RR*cxr 9527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-xr 9532
This theorem is referenced by:  iooss1  11445  limsupgord  13067  pnfnei  18955  dvfsumrlimge0  21634  dvfsumrlim2  21636  tanord1  22125  rlimcnp  22491  rlimcnp2  22492  dchrisum0lem2a  22898  pntleml  22992  pnt  22995
  Copyright terms: Public domain W3C validator